The attached abstract "Crossing into Uncharted Territory--The
Concept of Approximate X" is for your information and comments, if any.
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In science — and especially in mathematics — it is a universal practice
to express definitions in a language based on bivalent logic. Thus, if C
is a concept, then under its definition every object, u, is either an
instance of C or it is not, with no shades of gray allowed. This
deep-seated tradition — which is rooted in the principle of the excluded
middle—is in conflict with reality. Furthermore, it rules out the
possibility of graceful degradation, leading to counterintuitive
conclusions in the spirit of the ancient Greek sorites paradox.
In fuzzy logic — in contrast to bivalent logic — everything is, or is
allowed to be, a matter of degree. This is well known, but what is new
is the possibility of employing the recently developed fuzzy-logic-based
language PNL (Precisiated Natural Language) as a concept definition
language to formulate definitions of concepts of the form “approximate
X,” where X is a crisply defined bivalent-logic-based concept. For
example, if X is the concept of a linear system then “approximate X”
would be a system that is approximately linear.
The machinery of PNL provides a basis for a far-reaching project aimed
at associating with every or almost every crisply defined concept X a
PNL-based definition of “approximate X,” with the understanding that
“approximate X” is a fuzzy concept in this sense that every object x is
associated with the degree to which x fits X, with the degree taking
values in the unit interval or a partially ordered set. A crisp
definition of “approximate X” is not acceptable because it would have
the same problems as the crisp definition of X.
As a simple example, consider the concept of a linear system. Under the
usual definition of linearity, no physical system is linear. On the
other hand, every physical system may be very viewed as being
approximately linear to a degree. The question is: How can the degree be
defined?
More concretely, assume that I want to get a linear amplifier, A, and
that the deviation from linearity of A is described by the total
harmonic distortion, h, as a function of power output, P. For a given
h(P), then, the degree of linearity may be defined in the language of
fuzzy if-then rules – a language which is a sublanguage of PNL. In
effect, such a definition would associate with h(P) its grade of
membership in the fuzzy set of distortion/power functions which are
acceptable for my purposes. What is important to note is that the
definition would be local, or, equivalently, context-dependent, in the
sense of being tied to a particular application. What we see is that the
standard, crisp, definition of linearity is global (universal,
context-independent,), whereas the definition of approximate linearity
is local (context-dependent). This is a basic difference between a crisp
definition of X and PNL-based definition of “approximate X.” In effect,
the loss of universality is the price which has to be paid to define a
concept, C, in a way that enhances its rapport with reality.
In principle, with every crisply defined X we can associate a PNL-based
definition of “approximate X.” Among the basic concepts for which this
can be done are the concepts of stability, optimality, stationarity and
statistical independence. But a really intriguing possibility is to
formulate a PNL-based definition of “approximate theorem.” It is
conceivable that in many realistic settings assertions about
“approximate X” would of necessity have the form of “approximate
theorems,” rather than theorems in the usual sense. This is one of the
many basic issues which arise when we cross into the uncharted territory
of approximate concepts defined via PNL.
A simple example of “approximate theorem” is an approximate version of
Fermat’s theorem. More specifically, assume that the equality is
replaced with approximate equality. Furthermore, assume that x, y, z are
restricted to lie in the interval [I,N]. For a given n, the error, e(n),
is defined as the minimum of a normalized value of over all allowable
values of x, y, z. Observing the sequence {e(n)}, n = 3,4,…, we may form
perceptions described as, say, “for almost all n the error is small;” or
“the average error is small;” or whatever appears to have a high degree
of truth. Such perceptions, which in effect are summaries of the
behavior of e(n) as a function of n, may qualify to be called
“approximate Fermat’s theorems.” In a sense, an approximate theorem may
be viewed as a description of a perception. The concept of a fuzzy
theorem was mentioned in my 1975 paper “The Concept of a Linguistic
Variable and its Application to Approximate Reasoning.” What was missing
at the time was the concept of PNL.

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With my warm regards and best wishes for
the Christmas and Happy New Year

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